# recovering the information of the vertices from a simplex

It seems quite obvious that, when given a simplex, its set of vertices is uniquely determined by the simplex. The formal formulation of this intuition is as follows:

Suppose that the points $$\{v_0,v_1,\dots,v_k\}$$ and $$\{w_0,w_1,\dots,w_l\}$$ are affinely independent sets of points of $$\mathbb{R}^n$$. If the $$k$$-simplex $$\sigma=[v_0,\dots,v_k]$$ and the $$l$$-simplex $$\tau=[w_0,\dots, w_l]$$ are equal, then $$k=l$$ and $$\{v_0,v_1,\dots,v_k\}=\{w_0,w_1,\dots,w_l\}$$.

We clearly have $$k=l$$ because an $$m$$-simplex is homeomorphic to the closed ball of dimension $$m$$ and no two closed ball of different dimensions are homeomorphic to each other. But I cannot prove that $$\{v_0,v_1,\dots,v_k\}=\{w_0,w_1,\dots,w_l\}$$.

I think I must be overlooking something very obvious... Can anyone help me? Thanks in advance.

• What does it mean for two simplices to be equal? – Connor Malin Mar 27 '19 at 3:46
• @ConnorMalin Two simplices are equal when they are equal as sets. – Ken Mar 27 '19 at 4:04
• Given a simplex, you can recover its vertices by its extreme points. Maybe this helps. – JHF Mar 27 '19 at 16:33

Trickier than I expected. Suppose that $$\{v_0,\dots,v_k\}$$ does not equal $$\{w_0,\dots,w_k\}$$. Then without loss of generality we can assume $$v_0 = \Sigma r_i w_i$$ where multiple $$r_i$$ are nonzero. Again, without loss of generality assume that these are $$r_0,r_1$$. Then there is a function $$(-\epsilon, \epsilon) \rightarrow \sigma$$ defined by $$t \rightarrow (r_0 +t)w_0 + (r_1-t)w_1 + r_2 w_2 +\dots + r_k w_k$$. Such a function cannot exist because at $$t=0$$ this passes through $$v_0$$, and no such line segment passes through $$v_0$$ that is also contained in $$\sigma$$ because such a thing would necessarily contain points that when written as a sum of the $$v_i$$ would have negative coefficients.
As JHF had pointed out in the comment, the vertices of a given simplex can be characterized as the extreme points of the simplex. (If $$S$$ is a convex set, a point $$x$$ in $$S$$ is called its extreme point if for each $$y$$ in $$S\setminus \{x\}$$, any open line segment containing $$x$$ and $$y$$ always contains a point not in $$S$$.) This can be checked directly by using the barycentric coordinates.